Kunze Linear Algebra · Problem 10.2.14

Bilinear Forms

Mathematics — Methods · adjoint-matrix-dual-basis

Problem

Let

V

f

\mathfrak{B}

, and

\mathfrak{B}'

be as in Exercise 13. Suppose

T

is a linear operator on

V

and that

T'

is the operator which

f

associates with

T

as in Exercise 11. Show that (a)

[T']_{\mathfrak{B}'} = [T]_{\mathfrak{B}}^t

. (b)

\text{tr}\,(T) = \text{tr}\,(T') = \sum_i f(T\alpha_i, \alpha_i')